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SLIDE 1
Revision
Use long division to find
?
? How many times 23 goes into 8004
SLIDE 2
? ?
SLIDE 3
Dividing beyond the decimal point
Use long division to find as a decimal
?
?
SLIDE 4
? ?
SLIDE 5
Divide beyond the decimal point
? Go three times beyond the decimal point
? Go four times beyond the decimal point
What do you observe? Will you ever reach a definite answer?
SLIDE 6
? ?
We observe that in both cases the process repeats itself. We come back to the same calculation again and again and again. The result is an infinite decimal expansion.
SLIDE 7
One way in which we show that a number has an infinite decimal expansion is by using dots.
The repetition of a sequence of numbers makes it clear that the digit 1 and the digits 90 will be repeated infinitely in the two cases.
Use long division to find the infinite decimal expansion of and .
SLIDE 8
? ?
SLIDE 9
When we are using dots to show the infinitely repeating sequence of digits, we need to repeat that sequence at least twice. In the case of the extra 3 before the dots is not necessary. But representing infinitely repeating decimals in this way can be awkward, as the example of illustrates, where we must repeat the digits 142857 at least twice.
Infinitely repeating decimals may alternative be represented by a dot notation. A dot is placed over the first and last digits in the repeating sequence.
Task
Use the calculator to find . Express it as an infinitely recurring decimals using the dot notation.
SLIDE 10
SLIDE 11
? Express using the dot notation
? Express as an infinitely repeating decimals using trailing dots…
? Use the calculator to find each of the following and express the answer using the dot notation. .
? What pattern do you observe in the answers to the last question?
SLIDE 12
?
?
?
? In each case the repeating sequence of digits is the same as the numerator of a fraction.
SLIDE 13
Converting a recurring decimal to a fraction
Convert the following recurring decimals to fractions
? ? ?
SLIDE 14
?
?
?
SLIDE 15
? Find
? Find
SLIDE 16
?
?
SLIDE 17
The calculator has a square root button
Use this button to calculate . Write down all the digits displayed on your calculator. Do you observe a repeating sequence of numbers? Do you think that if the display were large enough there would be a repeating sequence of numbers?
SLIDE 18
Here the dots do not indicate an infinitely repeating sequence of digits, as no such sequence has been found. No repeating sequence can be observed.
There is no infinitely repeating sequence for .
SLIDE 19
Fill in the missing gaps
?
?
?
SLIDE 20
?
?
?
SLIDE 21
By dividing the figure into equal triangles prove that the area of pale blue square is half that of the surrounding square. Find the area of each square.
SLIDE 22
We divide the square into 8 identical triangles, all of which have the same area. Then the outside square is made of all 8 of these and has an area of 4 sq. units. The pale blue inside square is made of 4 of the triangles and has an area of 2 sq. units.
SLIDE 23
Four squares are nested inside each other as shown. If the area of A is 4 sq. units, what are the areas of B, C and D?
What is the ratio of the areas ?
SLIDE 24
The area of B is half the area of A. Area of B
The area C half the area of B. Area of C
The area of D is half the area of C. Area of D is
SLIDE 25
The area of square A is 4 sq. units, and the area of square B is 2 sq. units. The length of the side of A, is .
What is the length, b, of the square B?
SLIDE 26
Area of with side
Area of with side
This is because the area of a square is the product of the lengths of both its sides, which are equal.
For A,
For B,
SLIDE 27
Equivalent to 1
Fill in the missing gaps
? ? ?
? ? ?
SLIDE 28
? ? ?
? ? ?
SLIDE 29
? ?
? ?
? ?
?
SLIDE 30
? ?
? ?
? ?
?
SLIDE 31
Types of number
We have met the following types of number:
counting numbers, integers and fractions.
COUNTING NUMBERS
These are positive, whole numbers, 0, 1, 2, 3. There is no biggest counting number. Counting numbers are the answers to questions such as, “How many sheep has farmer Brown in his field?”
Which of the following are counting numbers?
SLIDE 32
13, 1 and 0 are counting numbers.
are negative integers.
is a fraction. is something else, that we shall discuss shortly.
INTEGERS
These are positive and negative whole numbers
These are used to measure the number of units along a line in two directions relative to an origin. Integers are the answers to questions such as, “How many degrees colder is it today than it was yesterday?” There is no smallest and no biggest integer.
Which of the following are integers?
SLIDE 33
From this list are integers. All the rest are fractions or square roots.
FRACTIONS
These are positive and negative ratios.
We will show later that they can be written in a single list. They take the form where a and b are integers. These are used to measure the ratio of one length to another, or the fraction of a whole. They answer questions such as, “How many times does the width of a rectangle divide into its length?” and “If I divide a cake equally into five parts, what is the fraction of one of those parts?” The answers to such questions are not always whole numbers or integers.
All counting numbers and integers are fractions.
Which of the following fractions are also integers?
SLIDE 34
From this list are also integers.
Because fractions are ratios and ratios are fractions, we also call fractions rational numbers. Here the word “rational” indicates that a ratio is involved. For example, .
? “All counting numbers are integers and all integers are rational numbers.” Express this statement as a Venn diagram.
Aside: the origin of the word “ratio” is in Latin where it indicated a calculation involving numbers. It can also mean “reasoning and understanding”. The word rational is used in two main senses: in mathematics to mean the type of number that comes from comparing one size to another; and more generally to indicate someone who has sound thinking or judgement.
SLIDE 35
SLIDE 36
Equivalent fractions
because
Two fractions are equivalent if there is an integer that when multiplied by the numerator and denominator of one fraction gives the numerator and denominator of the other.
Which one of the following is equivalent to ?
SLIDE 37
Of these numbers
because .
In the case of , a positive fraction, such as can never be equivalent to a negative fraction, and vice-versa. We must multiply one number by a negative number, here , and the other by a positive number, , and is not the same number as
SLIDE 38
? Express the lengths of the sides a and b as a ratio
? Express the ratio as a fraction
? Find the number that multiplies to give
? What happens if we multiply both the top and bottom of the fraction by ?
? Could there be a number that when we multiply the top and bottom of makes equivalent to a fraction?
(Here top means numerator and bottom means denominator. Top and bottom are often used as simpler alternatives to numerator and denominator of a fraction.)
SLIDE 39
?
?
?
?
? Could there be a number that when we multiply the top and bottom of makes equivalent to a fraction?
Answer: The only way in which we can turn on the bottom of into an integer is to multiply it by because . But to make the fraction equivalent we must multiply the top by the same number. Then has in it, and in the example the 2s cancel out. We can never get rid of a .
There can be no such number.
SLIDE 40
Irrational numbers
A number that cannot be equivalent to a fraction is called an irrational number.
The word “irrational” means “not a rational number”.
We have just proven that cannot be equivalent to a fraction, and, hence, is an irrational number.
The square root of any prime number is an irrational number.
Any square root of a square number is rational.
? Which of the following are prime numbers?
? Which of the following are square numbers?
? Classify each as rational or irrational
SLIDE 41
? are all prime numbers
? are all square numbers
?
From this list the square roots of the prime numbers are irrational
The square roots of squares are all rational
SLIDE 42
Roots of fractions
? Work out ? Work out
? Simplify ? Simplify
? Is a rational or an irrational number?
? Is a rational or an irrational number?
? Is a rational or an irrational number?
? Is a rational or an irrational number?
SLISE 43
? ?
? cannot be simplified further
? cannot be simplified further
? is an irrational number.
cannot be further simplified.
? is a rational number.
? is an irrational number.
cannot be further simplified.
? is an irrational number.
cannot be further simplified.
SLIDE 44
REAL NUMBERS
No rational number is an irrational number and vice-versa.
A real number is any number that is either rational or irrational.
All real numbers are either rational numbers or irrational numbers, and no number is both rational and irrational.
Express these statements in a Venn diagram.
SLIDE 45
All real numbers are either rational numbers or irrational numbers, and no number is both rational and irrational.
The collection of set of all real numbers is partitioned into the disjoint sets of rational and irrational numbers.
SLIDE 46
Fractions have either finite or infinitely repeating decimal expansions.
Since irrational numbers cannot be equivalent to a fraction, they have infinite non-repeating decimal expansions.
Use the calculator to compute . Write down all the digits displayed on your calculator. Will any sequence of digits ever repeat itself infinitely in the decimal expansion of ? Will there ever be an end to the decimal expansion of ?
SLIDE 47
No finite sequence of digits will ever repeat itself infinitely in the decimal expansion of . The expansion will go on forever.
SLIDE 48
Which of the following will have an infinite, non-repeating decimal expansion?
? ?
? ?
SLIDE 49
Which of the following will have an infinite, non-repeating decimal expansion?
This is the same as asking which of them is irrational
? rational, no
? irrational, yes
? irrational, yes
? rational, no
SLIDE 50
If we shorten or truncate this infinite expansion in any way, we obtain an approximation.
When approximating the rule is that if the digit behind the digit is 5 or greater, we round up.
is so the 3 rounds up.
? Using a calculator compute writing down all the digits on your display. Show that has an infinite decimal expansion.
? Approximate to 2, 4, 5 and 6 decimal places.
SLIDE 51
?
The dots show that the expansion is infinite, and we know that this is non-repeating, because 3 is a prime number.
?
to 4 decimal places. The 5 behind the 0 makes the 0 round up to a 1.
SLIDE 52
When we make an approximation, we do not write “to 5 decimal places”, we abbreviate this to .
? Find
? Approximate to
SLIDE 53
?
?
SLIDE 54
An approximation by a decimal expansion is not exact.
This is true. But is not true (false), because has an infinite decimal expansion.
However, the symbol tells us exactly what is. Thus,
is exact and is inexact.
is an inexact approximation.
? . Is 3.61 exact or inexact?
? Which of the following are exact?
? Is the statement true or false?
? Is the statement true or false? Is 4.5386 exact or inexact?
SLIDE 55
? In , 3.61 is inexact.
?
All of these are exact.
? is false. It states that is equal (exactly) to 4.1231, and is only approximately equal to this number. So, it is false.
? is true because an approximation sign has been used. It does not state a falsehood that is equal to 4.5386, and it also states correctly the degree of approximation as 4 d. p. in the number 4.5386 is an inexact approximation to .
SLIDE 56
The calculator computes but how could this number be found?
How do we compute ?
? What is ? What is ?
? Use a calculator to find . Is this number bigger or smaller than 2?
? Use a calculator to find . Is this number bigger or smaller than 2?
? How do we know that ?
SLIDE 57
? and
? and
?
? We know that because
SLIDE 58
Each of the above diagrams represents an interval. In these diagrams an open circle represents an end-point that is not included in the interval, and a filled circle represents an end-point that is included in the interval. Each of the above corresponds to an inequality.
? ? ? ?
SLIDE 59
Find inequalities to represent each of these intervals.
SLIDE 60
? ? ? ?
SLIDE 61
The method of bisection
We know that because
is an approximation for .
We can improve upon this approximation by repeating the process.
In the above diagram we have divided the interval into two equal halves. This is called bisection.
To improve the approximation, we ask in which of these two halves does lie?
Since we already know that and we only need to know what is.
Find and determine in which of these two intervals lies. Find an improved approximation for by this means.
SLIDE 62
because
The mid-point between 1.4 and 1.45 is the average of these points
SLIDE 63
Iteration
To repeat the same process again and again is called iteration
because
Iterate (repeat) this process of bisection one more time to obtain an improved approximation
SLIDE 64
because
SLIDE 65
After three iterations of the method of bisection we have trapped within the interval
The width of this interval is 0.025.
How many more iterations are needed to have an interval of width ?
SLIDE 66
The width of the interval is always half the width of the previous interval
We must repeat this process two more times to obtain an approximation to within 0.01.
SLIDE 67
Trial and improvement
The method of bisection will find a square root to any degree of required accuracy, but it finds such approximations somewhat slowly.
Trial and improvement is a similar method that uses educated guesswork to improve the speed of convergence.
We use the information given in the process to choose an interval that is more likely to give quicker convergence.
Example
Find to 3 d.p.
SLIDE 68
Find to 3 d.p.
SLIDE 69
SLIDE 70
Find to 3 d.p.
SLIDE 71
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